**Problem 1:**

Axl Rose, the lead singer of Guns N’ Roses, needs your help to invest his money to pay for his son’s university education. His son, Dylan, is considering entering an Ivy League university 5 years from now. Axl Rose can only give his money to you now and wants to minimize his initial investment. You can put his money in more than 1 investment type. No new money for investment in the second year and onward because he is busy with a worldwide tour with his music group and a U.K. rock band Duran Duran. Any new investment after the first year can only

be made through reinvestment of previous investment.

Dylan’s first year university expenses will amount to $70,000 and go up by $4,000 per year in the last three years of his education. Assume that now is the beginning of year 1. It means that Axl Rose will withdraw his investment at the beginning of years 6, 7, 8, and 9. Some possible investments:

**a.** Visualize where you want to finish. What numbers are needed? What are the decisions that need to be made? What should the objective be?

**b.** Create an LP model in algebraic form

**c.** Find the optimal solution using Solver.

**d.** Explain the optimal solution. Example: Axl Rose should invest $X in P at the beginning of the first year, keep his investment in P for 2 years, and then receive $Y from divesting investment P at the beginning of the third year. He then should invest $Y at Q at then beginning of the third year …

**Problem 2:**

Essence Beverage Co. produces a variety of fruit drinks. The company will introduce a new blend of strawberry, blackberry, and apple juices. The new blend must consist of at least 50% of strawberry juice, between 20% – 30% of blackberry juice, and exactly 20% of apple juice. For then current session, 10,000 gallons of strawberry juice and 8,500 gallons of blackberry juice can be purchased. Apple juice is unlimited. The costs are $1.00 per gallon for the strawberry juice, $1.50 per gallon for the blackberry juice, and $0.50 per gallon for the apple juice. The company can sell the entire production it can blend for $2.5 per gallon.

**a.** Create an LP model to maximize the total profit in algebraic form.

**b.** Find the optimal solution using Solver. Use the sensitivity analysis to answer the next questions.

**c.** Refer to the “Variables Cells” in the sensitivity analysis. Explain each of the final values, reduced costs (use Google), and objective coefficients. Calculate and interpret the allowable range of each objective coefficient.

**d.** Refer to the “Constraints” in the sensitivity analysis. Explain each of the final values, shadow prices, and constraint R.H. sides. Calculate and interpret the allowable range of each constraint R.H. side. What are the managerial implications of the shadow prices?

**e.** If the company could get extra amounts of the strawberry juice, should the company do so? If so, how much should the company willing to pay for each extra gallon of the strawberry juice? How many extra gallons would the company want to acquire?

**f.** If the company could acquire extra amounts of the blackberry juice, should the company do so? If so, how much should the company willing to pay for each extra gallon of the blackberry juice? How many extra gallons would they want to buy?

**Problem 3:**

Red Wolf Manufacturing Co. has 3 factories, 2 distribution centers (DCs), and 4 major customers. Maximum factory capacities in units and shipping cost per unit in $ from each factory to each DC are as follows:

Customer demand in units and shipping cost per unit in $ from each DC to each customer are as follows:

Red Wolf must satisfy all demands. It is acceptable if some factories do not produce at the maximum capacity.

**a.** (1) Draw a network that depicts the company’s supply chain network. Identify the supply nodes, transshipment nodes, and demand nodes in this network. Handwritten submissions/pictures are not acceptable. Use Excel, Word, or any other software. A simple picture is enough.

(2) Formulate this problem in algebraic form.

(3) Solve this problem using Solver.

**b.** A new supply chain manager just arrived. His name is Bono. He just retired as the lead singer of U2. He reorganized the logistics system and decided that freights between the two DCs are allowed at $5 per unit and that direct deliveries can be made from FC to DZ at a cost of $35 per unit.

(1) Draw a network that depicts the new company’s supply chain network. Identify the supply nodes, transshipment nodes, and demand nodes in this network.

(2) Formulate this problem in algebraic form.

(3) Solve this problem using Solver. Compare your solutions in parts (a) and (b).

(4) What managerial lessons can be learned from this problem?

**Problem 4:**

NorthStar Bank is expanding into a state that has 13 counties. State archaic law allows creating bank branches in any counties that are adjacent to a county in which a principal place of business (PPB) is situated. Hence, the bank serves the people in a county where the bank sets up a PBB and any adjacent counties.

**a.** Suppose that only 1 PBB can be set up in the state. Where should the PBB be situated to maximize the number of people served? Formulate this problem in algebraic form and then solve this problem using Solver.

**b.** If 2 PBBs can be established in the state, where should the 2 PBBs be located? Formulate this problem in algebraic form and then solve this problem using Solver.

**c.** North Star Bank management has learned that a local bank in county 5 is for sale. If only 1 PBB can be established in the state, should the management buy the local bank? Justify your answer with a calculation, i.e., the number of people served. Compare your calculations in parts (a) and (c).

**d.** Suppose NorthStar Bank has decided to buy the local bank in county 5 because the local bank is cheap. If 2 PBBs can be established in the state and one of them in county 5, where should the other PBB be located to maximize the number of people served? Formulate this problem in algebraic form and then solve this problem using Solver.

**e.** What managerial lessons have you learned from this problem?

The map displays the state that has 13 counties with the number of people in each county.

**Problem 5:**

Sky Gate Co. tests and repairs high-tech equipment. Jobs arrive at the rate of 3 jobs/day. One day= 8 hours. The interarrival times follow an exponential distribution. The company’s testing facility is a single-channel system operated by a crew of 4 highly skilled technicians. The service times have a mean of 2 hours and follow an exponential distribution. The testing costs the company $285 per hour. The waiting cost is $360 per hour.

**a.** What is the proper waiting line model for this problem? M/D/3? M/G/4? Or something else? Explain.

**b.** Find and explain the lambda, mu, s, L, Lq, W, Wq, rho, Po, and total cost of this particular model.

**c.** The crew leader suggested the company management to buy a new automated testing equipment that would allow a constant testing time of 1.5 hours/job. Assume the standard deviation is 0. However, the new testing cost, including equipment amortization, would be $330 per hour. What is the proper waiting line model for this new equipment? Find and explain the new lambda, mu, s, L, Lq, W, Wq, rho, Po, and total cost. Should the company buy the new equipment?

**d.** If the new equipment only needs 3 technicians, what should the company do with the other technician? Use your management knowledge and Google.

**e.** Is high process variation good or bad for most companies? What are the effects of reducing process variation on L, Lq, W and Wq? What management techniques are useful to reduce process variation? Use Google. We also discussed this topic in class.

**f.** Is it reasonable to think that companies such as SkyGate can attract new customers by reducing waiting time? Create and explain a table for this problem similar to table 12.2 on page 508.

**g.** What are the managerial lessons of this problem?

PS: If you use Google, don’t copy and paste. Use your own words. List all your references including web site addresses.

**Problem 6:**

Oasis Business Advisory, a consulting firm, leases 1 copy machine for $50 per day that is used by all consultants. Six consultants use the machine per hour. Average usage (service time) = 9 minutes/consultant. The interarrival and service times follow the exponential distribution.

**a.** What is the proper waiting line model for this problem? M/D/5? M/G/8? Or something else? Explain.

**b.** How do you know that 1 copy machine can serve 6 consultants/hour?

**c.** Use chapters 11 (Excel template) and 12 (queuing simulator) to find and explain L, Lq, W, Wq, and Po. Do you get similar results?

**d.** Suppose that the average salary (waiting cost) is $25/hour for each consultant. What is the total cost/day of having 1 copy machine? 1 Day = 8 working hours.

**e.** What is the optimum number of copy machines to minimize the total cost/day? Justify your answer. Are there any other factors, besides the total cost, that should be considered before leasing more than 1 copy machine?

**f.** What are the similarities and differences between the techniques in chapter 11 (analytical methods of waiting line models) and those in chapter 12 (simulation of waiting line models)? Which technique do you prefer to analyze a simple problem? If you have to analyze a complex network of machines such as a factory that has 100 machines, which technique would you use? You can also Google “process simulation” to enrich your answer.

**g.** What managerial lessons do you learn from this problem?

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